We assume now that the sequence \(\nu =(\nu ^{(l)}(X))_{l = 0}^L\)  of outputs of the hidden layers of our SDNN is a martingale; this allows us to state some interesting properties of SDNNs. Note that if \(\nu \)  is any kind of martingale (very-weak, weak, or strong, see Appendix ), then the width
of the layers of the SDNN is constant throughout the neural network and equal to \(p\) . Indeed, if \(n_l=n_{l-1}=p\)  does not hold for all \(l \in \lbrace 0,\ldots ,L\rbrace \) , then computing \(\mathbb {E}[\nu ^{(l)}(X) \mid \nu ^{(0)}(X),\ldots ,\nu ^{(l-1)}(X)]\)  may not have a clear mathematical
meaning. Neural networks with
constant width are used extensively and a well studied subject [1]}.
